275
Scientiae Mathematicae Japonicae Online, e-2012, 275–284
NAZAROV TYPE UNCERTAINTY INEQUALITY
FOR FOURIER SERIES
A. Abouelaz and T. Kawazoe
Received July 4, 2012; revised July 5, 2012
Abstract. We shall obtain an analogue of Nazarov’s uncertainty inequality for ndimensional Fourier series from the one for n-dimensional Fourier transform. Some
inequalities are new and better than ones deduced from a classical local uncertainty
inequality.
1 Introduction The uncertainty principle generally asserts that a non-zero function and
its Fourier transform cannot be too small simultaneously. Although interpretations of the
smallness are very broad, various versions of the uncertainty principle exist and also, analogous versions for Fourier series exist. However, as for the Nazarov uncertainty principle, it
is known only for Fourier transform. In this paper we shall obtain an analogous uncertainty
inequality for n-dimensional Fourier series. Nazarov’s uncertainty inequality is originally
appeared in [2] and P. Jaming [1] extends it to a higher dimensional Fourier transform:
There exists a constant C such that, if S, Σ are subset of Rn of finite measure, then for
every f ∈ L2 (Rn ),
∫
(1)
Rn
|f (x)|2 dx ≤ γ(S, Σ)
(∫
∫
Rn \S
|f (x)|2 dx +
Rn \Σ
)
|Ff (x)|2 dx ,
where
1
1
γ(S, Σ) = CeC min{|S||Σ|,|S| n w(Σ),|Σ| n w(S)} ,
Ff is the Fourier transform of f , | · | is the measure and w(·) is the mean width. We shall
rewrite this inequality for a Fourier series version. For f ∈ l1 (Zn ) ∩ l2 (Zn ), we denote by fˇ
the trigonometric series
∑
fˇ(λ) =
f (m)e2πiλm , λ ∈ Rn ,
m∈Zn
and for any function g ∈ Lp (Rn ), 1 ≤ p ≤ ∞, we denote by f £ g a convolution
f £ g(x) =
∑
f (m)g(x − m), x ∈ Rn .
m∈Zn
Applying the above inequality (1) to the function f £ g on Rn , we can deduce a Nazarov
type uncertainty inequality for f on Tn . Let I be a closed measurable subset in [− 12 , 12 ]n ,
S a finite subset in Zn and Σ a measurable subset in I. Then for all functions gI ∈ L2 (Rn )
2000 Mathematics Subject Classification. 42B05; 26D05.
Key words and phrases. Nazarov’s uncertainty principle; Fourier series.
276
A. ABOUELAZ AND T. KAWAZOE
supported on I and vanishing on the boundary of I, the following inequality follows (see
Theorem 3.4 and let J = {0}):
∑
|f (m)|2
m∈Zn
≤γ(S + I, Σ)
( ∑
∫
|f (m)|2 +
m∈Zn \S
)
(
|F(gI )(x)|2 )
|fˇ(x)|2 1 −
dx
.
kgI k2L2 (Rn )
Σ
∫
Tn \Σ
|fˇ(x)|2 dx +
Especially, letting I = [− 12 , 12 ]n and taking the characteristic function χ of [− 12 , 12 ]n as gI ,
we see that
1−
n ¯
¯
∏
|F(χ)(x)|2
¯ sin πxi ¯2
=
1
−
¯
¯ = O(|x|2 ), x = (x1 , x2 , · · · , xn ),
2
kχkL2 (Rn )
πx
i
i=1
and therefore, we can deduce that
(2)
∑
|f (m)| ≤ γ(S + I, Σ)
2
(
∑
∫
|f (m)| +
2
m∈Zn \S
m∈Zn
Tn \Σ
∫
|fˇ(x)|2 dx + C
|fˇ(x)|2 |x|2 dx
)
Σ
(see §4, (I)). Clearly, if we can find a function gI satisfying
|F(gI )(x)|2
= 1 + O(xγ ),
kgI k2L2 (Rn )
then we can replace |x|2 in the last integral of (2) by |x|γ . However, we see that γ = 2
associated to χ is maximal when I is a subset in [− 12 , 12 ]n (see Remark 4.1). Therefore, in
§3 we shall consider a general subset I ⊂ Rn and obtain a modified inequality of (2). In
§4, we find an example of the modified inequality where |x|2 in (2) is replaced by |x|10 (see
Corollary 4.2).
On the other hand, J. F. Price and P. C. Racki [3] obtain the so-called local uncertainty
inequalities for Fourier series. As a special case, for β < 21 , α ≥ 0 and α ≥ β, there exists a
constant K such that for all fˇ ∈ L1 (Tn ) and all finite S of Zn ,
∫
∑
2
2
2α
|f (m)| ≤ K |S|
|fˇ(x)|2 |x|2nβ dx.
Tn
m∈S
Hence, it easily follows that
(3)
∑
m∈Zn
|f (m)|2 ≤
∑
m∈Zn \S
∫
∫
|f (m)|2 + K 2 |S|2α
Tn \Σ
|fˇ(x)|2 |x|2nβ dx.
|fˇ(x)|2 dx + K 2 |S|2α
Σ
Here the weight function |x|2nβ , which reflects the localization of fˇ around zero, is better
than |x|2 in (2) when n ≥ 3. In our modified inequalities (see Theorem 3.4 and Remark
3.5), there is some possibility of replacing |x|2nβ by |x|γ . At this stage, γ = 10 is our best
result (see §4, (IV)).
by Lp (Tn ) the space of all
2 Notations. We identify [− 12 , 12 ]n with Tn . We denote
∫
p
n
measurable functions f (t) on T such that kf kLp (Tn ) = Tn |f (t)|p dt < ∞ for 1 ≤ p < ∞
and kf kL∞ (Tn ) = ess.supt∈Tn |f (t)| < ∞ for p = ∞. Similarly, we denote by lp (Zn ) the space
NAZAROV TYPE UNCERTAINTY INEQUALITY FOR FOURIER SERIES
277
∑
of all functions F (m) on Zn such that kF kplp (Zn ) = m∈Zn |F (m)|p < ∞ for 1 ≤ p < ∞ and
kF kL∞ (Zn ) = supm∈Zn |F (m)| < ∞ for p = ∞. For f ∈ L1 (Tn ), the Fourier coefficients of
f are defined by
∫
fˆ(m) =
f (t)e−2πitm dt, m ∈ Zn .
Tn
and kfˆkL∞ (Zn ) ≤ kf kL1 (Tn ) . For F ∈ l1 (Zn ), we denote by F̌ the trigonometric series
∑
F̌ (λ) =
F (m)e2πiλm , λ ∈ Rn .
m∈Zn
F̌ is a 1-periodic function on Rn that can be regarded as a function on Tn and kF̌ kL∞ (Tn ) ≤
ˇ
kF kL1 (Zn ) . If f and fˆ belong to L1 (Tn ) and l1 (Zn ) respectively, then f = fˆ. Moreover, if
f ∈ L2 (Tn ), then fˆ ∈ l2 (Zn ) and kf kL2 (Tn ) = kfˆkL2 (Zn ) . We define a convolution ⊗ on Zn
as follows: For F, G ∈ l1 (Zn ),
∑
F ⊗ G(n) =
F (m)G(n − m).
m∈Zn
Then F ⊗ G ∈ l1 (Zn ) and (F ⊗ G)ˇ = F̌ Ǧ.
p
n
We denote
) the space of all measurable functions f (x) on Rn such that
∫ by L (R
p
p
kf kLp (Rn ) = Rn |f (x)| dx < ∞ for 1 ≤ p < ∞ and kf kL∞ (Rn ) = ess.supx∈Rn |f (x)| < ∞ for
p = ∞. For f ∈ L1 (Rn ), we denote the Fourier transform of f as
∫
F(f )(λ) =
f (x)e−2πiλx dx, λ ∈ Rn .
Rn
Clearly kF(f )kL∞ (Rn ) ≤ kf kL1 (Rn ) . For F ∈ L1 (R), the inverse Fourier transform F −1 is
given by
∫
1
F −1 (F )(x) =
F (λ)e2πiλx dλ, x ∈ Rn .
(2π)n Rn
If f and F(f ) ∈ L1 (Rn ), then f = F −1 (F(f )). Moreover, if f ∈ L2 (Rn ), then F(f ) ∈ l2 (Rn )
n
and (2π) 2 kf kL2 (Rn ) = kF(f )kL2 (Rn ) . We define a convolution ∗ on Rn as follows: For
f, g ∈ L1 (Rn ),
∫
f ∗ g(x) =
f (y)g(x − y)dy.
Rn
Then f ∗ g ∈ L1 (R1 ) and F(f ∗ g) = F(f )F(g).
For f ∈ l1 (Zn ) and g ∈ Lp (Rn ), we define a convolution £ as
∑
f (m)g(x − m), x ∈ Rn .
f £ g(x) =
m∈Zn
We can easily see that for 1 ≤ p ≤ ∞,
kf £ gkLp (Rn ) ≤ kf kl1 (Zn ) kgkLp (Rn )
and, if p = 1, then
(4)
F(f £ g)(λ) = fˇ(λ)F(g)(λ).
278
A. ABOUELAZ AND T. KAWAZOE
3 Nazarov’s inequality on Zn . Let I be a connected subset of Rn of finite measure
and containing 0 and put
2I ◦ ∩ Zn = J,
where I ◦ = I − ∂I. Let J ] denote the cardinal of J. For a finite subset S of Zn , let S̃ be a
subset of Rn defined by
S̃ = S + I.
Then (S + 2I ◦ ) ∩ Zn = S + J. In what follows gI is a function in L2 (Rn ) which is supported
on I and vanishes on ∂I. We define
∫
2
kgI k2,j =
gI (x)gI (x − j)dx
Rn
for j ∈ J.
Lemma 3.1 For f ∈ l1 (Zn ) ∩ l2 (Zn ), it follows that
∑
∑
kf £ gI kL2 (Rn ) =
kgI k22,j
f (m)f (m + j)
m∈Zn
j∈J
(5)
=
∑
∫
kgI k22,j
Tn
j∈J
|fˇ(λ)|2 e2πijλ dλ ≤ J ] kgI k2L2 (Rn ) kf k2l2 (Zn ) .
Proof. Since gI (x) = 0 if x ∈
/ I ◦ and 2I ◦ ∩ Zn = J, it follows that
∫
∑ ∑
kf £ gI k2L2 (Rn ) =
f (m)f (m0 )
gI (x − m)gI (x − m0 )dx
m∈Zn m0 ∈Zn
=
=
∑
∑
m∈Zn
m0 ∈Zn
∑ ∑
∫
Rn
f (m)f (m0 )
Rn
gI (x)gI (x − (m0 − m))dx
∫
f (m)f (m + j)
j∈J m∈Zn
Rn
gI (x)gI (x − j)dx.
The equality (5) follows from the Parseval equality for Fourier series. The inequality is
obvious from the Schwartz inequality.
¤
Lemma 3.2 Let f ∈ l1 (Zn ) and gI , S, S̃ be as above. Then for all x ∈ Rn ,
(6)
|f | £ |gI |(x)χS̃ (x) ≤ |f χS+J | £ |gI |(x) ≤ |f | £ |gI |(x)|χS̃+J (x),
where χS+J and χS̃ , χS̃+J are the characteristic functions of the sets S + J ⊂ Zn and
S̃, S̃ + J ⊂ Rn respectively. Especially, when J = {0}, it follows that
(7)
f £ gI (x)χS̃ (x) = (f χS ) £ gI (x).
Proof. We recall that |f | £ |gI |(x) is given by
∑
|f (m)||gI (x − m)|.
m∈Zn
NAZAROV TYPE UNCERTAINTY INEQUALITY FOR FOURIER SERIES
279
We suppose that χS̃ (x) = 1 and g(x − m) 6= 0. Then m ∈ x − I ◦ ⊂ S̃ + I ◦ = S + 2I ◦ .
Since (S + 2I ◦ ) ∩ Zn = S + J, it follows m ∈ S + J and χS+J (m) = 1. If χS+J (m) = 1 and
gI (x − m) 6= 0, then x ∈ m + I ⊂ S + J + I = S̃ + J and χS̃+J (x) = 1. Hence the desired
inequalities (6) follow. Let J = {0} and suppose that g(x − m) 6= 0. Then it follows that
χS̃ (x) = 1 if and only if χS+J (m) = 1. Hence we can deduce the equality (7).
¤
Corollary 3.3 Let I, J, gI , S, S̃ be as above and Σ a measurable subset in Rn such that
Σ ⊂ Tn . Then for f ∈ l1 (Zn ),
∫
∑
(i)
|f £ gI (x)|2 dx ≤ J ] kgI k2L2 (Rn )
|f (m)|2
Rn \(S̃+J)
m∈Zn \(S+J)
and the equality holds when J = {0}.
∫
∫
∑
2
2
(ii)
|F(f £ gI )(x)| dx =
kgI k2,j
Rn \Σ
Tn \Σ
j∈J
∫
|fˇ(λ)|2 e2πijλ dλ
∑
|fˇ(λ)|2 (
kgI k22,j e2πijλ − |F(gI )(λ)|2 )dx.
+
Σ
j∈J
Proof. (i) : By using Lemma 3.1 and Lemma 3.2, we see that
∫
|f £ gI (x)|2 dx ≤ k|f | £ |gI |k2L2 (Rn ) − k|f | £ |gI · χS̃+J |k2L2 (Rn )
Rn \(S̃+J)
≤ k|f | £ |gI |k2L2 (Rn ) − k|f χS+J | £ |gI |k2L2 (Rn )
= k|f (1 − χS+J )| £ |gI |k2L2 (Rn )
∑
|f (m)|2 .
≤ J ] kgI k2L2 (Rn )
m∈Zn \(S+J)
When J = {0}, the equalities (5) and (7) yield the desired equality.
(ii) : It follows from (4) and Lemma 3.1 that
∫
|F(f £ gI )(x)|2 dx
Rn \Σ
∫
∫
2
=
|F(f £ gI )(x)| dx −
|F(f £ gI )(x)|2 dx
Rn
Σ
∫
∫
∑
=
|f £ gI (x)|2 dx −
kgk22,j
|fˇ(λ)|2 e2πijλ dλ
Rn
+
∑
=
j∈J
kgk22,j
∫
Σ
∫
|fˇ(λ)|2 e2πijλ dλ −
kgk22,j
j∈J
∑
j∈J
∫
Σ
Tn \Σ
|fˇ(λ)F(gI )(λ)|2 dx
∫
Σ
∑
|fˇ(x)|2 (
kgk22,j e2πijλ −|F(gI )(λ)|2 )dλ.
|fˇ(λ)|2 e2πijλ dλ+
Σ
j∈J
¤
Now applying Nazarov’s inequality (1) to f £gI and combining Lemma 3.1 and Corollary
3.3, we can deduce the following.
280
A. ABOUELAZ AND T. KAWAZOE
Theorem 3.4 Let I be a connected subset of Rn of finite measure and containing 0 and
put 2I ◦ ∩ Zn = J. Let S be a finite subset of Zn and Σ a measurable subset in Rn such that
Σ ⊂ Tn . Then for all f ∈ l1 (Zn ) ∩ l2 (Zn ),
∑
∑
kgI k2,j
f (m)f (m + j)
m∈Zn
j∈J
(
(
≤ γ(S̃ + J, Σ) J ] kgI k2L2 (Rn )
∫
∑
|f (m)| +
m∈Zn \(S+J)
∫
|fˇ(λ)|2
+
Σ
(∑
|fˇ(λ)|2 dλ
2
kgI k22,j e2πijλ
Tn \Σ
− |F(gI )(λ)|
2
)
)
)
dλ ,
j∈J
where γ(S̃ + J, Σ) is the Nazarov constant.
Remark 3.5 Let p ∈ Z and I = [−p − 21 , p + 12 ]n . Then Jpn = 2I ◦ ∩ Zn = {j =
(j1 , j2 , · · · , jn ) | −2p ≤ ji ≤ 2p, 1 ≤ i ≤ n} is a representative set of Zn /(2p + 1)Zn .
For each j ∈ Jpn we put
{
f (m) if m ∈ j + (2p + 1)Zn
.
fj (m) =
0
otherwise
∑
We note that f = j∈Jpn fj and apply the above theorem to each fj . Then summing up
each inequality, we can deduce the following:
∑
|f (m)|2
m∈Zn
(
(
≤γ(S̃ + Jpn , Σ) (2p + 1)n kgI k2L2 (Rn )
∫
+
∑
Σ j∈J n
p
|fˇj (λ)|2
(∑
∫
∑
|f (m)|2 +
m∈Zn \(S+Jpn )
kgI k22,j e2πijλ
∑
Tn \Σ j∈J n
p
− |F(gI )(λ)|
2
)
|fˇj (λ)|2 dλ
)
)
dλ .
j∈Jpn
4 Examples We here give some examples of gI and calculate explicitly the inequality in
Theorem 3.4. Especially, the order of the weight function in the second integral is important
for the localization of fˇ.
(I) First we shall consider the case of n = 1. Let I = [− 12 , 21 ] and gI be the characteristic
function of I:
{
1 |x| < 12
gI (x) = χ(x) =
0 |x| ≥ 12 .
Then J = {0} and kgI k2,0 = kgI k2L2 (R) = 1. Moreover, F(gI )(λ) =
O(|λ|4 ). Hence, it follows that
kgI k22,0 − |F(gI )(λ)|2 =
sin πλ
πλ
= 1−
(πλ)2
6
+
(πλ)2
+ O(|λ|4 ).
3
When n > 1, let I = [− 12 , 12 ]n and gI (x) = χ(x1 )χ(x2 ) · · · χ(xn ). Then it is easy to see that
kgI k22,0 − |F(gI )(λ)|2 = 1 −
n ¯
¯
∏
(π|λ|)2
¯ sin πλi ¯2
+ O(|λ|4 ).
¯
¯ =
πλ
3
i
i=1
NAZAROV TYPE UNCERTAINTY INEQUALITY FOR FOURIER SERIES
281
Remark 4.1 Let f be a function supported on I0 ⊂ I = [− 12 , 12 ]n and satisfy
kf k2L2 (Rn ) − |F(f )(0)|2 = 0.
(8)
Without loss of generality, we may suppose that kf k∞ ≤ 1. Since
¯ ∫
¯∫
¯
¯
f (x)dx¯ ≤
|f (x)|dx ≤ |I0 |1/2 kf kL2 (Rn ) ,
|F(f )(0)| = ¯
I0
I0
it follows that I = I0 and f (x) = 1 for all x ∈ I0 . Therefore, functions f satisfying (8) are
constant multiples of the characteristic function χ of I.
In the following we shall give some examples of gI , whose support is larger than [− 12 , 12 ]n .
We shall consider the case of n = 1, because, similarly as in (I), we can obtain a general
case from their products.
(II) Let n = 1, I = [−1, 1] and
gI = χ ∗ χ.
Then J = {−1, 0, 1} and
kgI k22,0 =
2
1
, kgI k22,±1 = .
3
6
Hence, it follows that
∑
kgI k22,j e2πijλ =
j∈J
2 1
2
2
+ cos 2πλ = 1 − (πλ)2 + (πλ)4 + O(|λ|6 ).
3 3
3
9
On the other hand,
F(gI )(λ)2 =
( sin πλ )4
πλ
2
1
= 1 − (πλ)2 + (πλ)4 + O(|λ|6 ).
3
5
Hence, it follows that
∑
kgI k22,j e2πijλ − |F(gI )(λ)|2 =
j∈J
(πλ)4
+ O(|λ|6 ).
45
(III) Let n = 1, Ip = [− 12 − p, 12 + p] and
gIp = χ(x − p) + χ(x + p) − 2χ(x)
for p = 1, 2, · · · . Then Jp = {−2p, −2p + 1, · · · , 2p − 1, 2p} and
kgIp k22,0 = 6,
{
−4 j = p
2
kgIp k2,±j =
0
j=
6 p


j = p2
1
2
kgIp k2,±j = −4 j = p


0
j 6= p2 , p
if j ≡ 1, 3 (mod 4),
if j ≡ 0, 2 (mod 4).
282
A. ABOUELAZ AND T. KAWAZOE
Hence, it follows that
∑
kgIp k22,j e2πijλ = 16(pπλ)4 +
j∈Jp
32
(pπλ)6 + O(|λ|8 ).
3
On the other hand,
F(gIp )(λ)2 =
( 2e−i2pπλ (−1 + ei2pπλ )2 sin πλ )2
2πλ
16 4
= 16(pπλ) + (p + 2p6 )(πλ)6 + O(|λ|8 ).
3
4
Therefore, it follows that
∑
kgIp k22,j e2πijλ − |F(gIp )(λ)|2 =
j∈Jp
16p4 (πλ)6
+ O(|λ|8 ).
3
(IV) Let n = 1, Ip = [−p − 12 , p + 12 ] and
gIp,q = (χ(x − p) + χ(x + p) − 2χ(x)) −
p2
(χ(x − q) + χ(x + q) − 2χ(x)),
q2
where p > 4q, p + q is even, q ≥ 2. Then Jp = {j | −2p ≤ j ≤ 2p} and
kgIp,q k22,0 = 2 +
kgIp,q k22,±2q =
2p4
q4
( 2
)2
+ 4 pq2 − 1 ,
p4
q4 ,
2
kgIp,q k22,±(p−q) = − 2p
q2 ,
2
kgIp,q k22,±(p+q) = − 2p
q2 ,
2
kgIp,q k22,±q = − 4p
q2
(
p2
q2
)
−1 ,
( 2
)
kgIp,q k22,±p = 4 pq2 − 1 ,
kgIp,q k22,±2p = 1
and kgIp,q k22,±j = 0 otherwise. Hence, it follows that
∑
kgIp,q k22,j e2πijλ
j∈J
16
64 4 2
= p4 (p2 − q 2 )2 (πλ)8 −
p (p − q 2 )2 (p2 + q 2 )(πλ)10 + O(|λ|12 ).
9
135
On the other hand,
(
sin πλ )2
F(gIp,q )(λ)2 = e−2π(p+q)iλ
2πq 2 λ
× (−q 2 e2πqiλ − q 2 e2π(2p+q)iλ + p2 e2πpiλ + p2 e2π(2q+p)iλ −2(p2 − q 2 )e2π(p+q)iλ )2
16 4 2
16 4 2
=
p (p − q 2 )2 (πλ)8 −
p (p − q 2 )2 (5 + 4p2 + 4q 2 )(πλ)10 + O(|λ|12 ).
9
135
Therefore, it follows that
∑
j∈J
kgIp,q k22,j e2πijλ − |F(gIp,q )(λ)|2 =
16 4 2
p (p − q 2 )(πλ)10 + O(|λ|12 ).
27
NAZAROV TYPE UNCERTAINTY INEQUALITY FOR FOURIER SERIES
283
At this stage, this |λ|10 is our best weight function. When n > 1, as said before, let
I = [−p − 12 , p + 12 ]n and gI (x) = gIp,q (x1 )gIp,q (x2 ) · · · gIp,q (xn ). Then J = Jpn = {j =
(j1 , j2 , · · · , jn ) ∈ Zn | −2p ≤ ji ≤ 2p, 1 ≤ i ≤ n} and it is easy to see that kgI k22,j =
∏n
i=1 kgIp,q k2,ji and
∑
kgI k22,j e2πijλ =
n
∏
∑
(
kgIp,q k2,ji e2πiji λi ).
i=1 ji ∈Jp
j∈Jpn
Since F(gI )(λ) = F(gIp,q )(λ1 )F(gIp,q )(λ2 ) · · · F(gIp,q )(λn ) and kF(gIp,q )k∞ ≤ kF(gIp,q )k1
=
4p2
q2 ,
it follows that
¯
¯∑
¯
¯
kgI k22,j e2πijλ − |F(gI )(λ)|2 ¯
¯
j∈Jpn
( 4p2 )n−1 16
≤ 2
p4 (p2 − q 2 )(|λ1 |10 + · · · + |λn |10 ) + O(|λ|12 ).
q
27
Corollary 4.2 Let S be a finite subset in Zn and Σ a measurable subset in [− 12 , 21 ]n . For
p > 4q, p+q even and q ≥ 2, let Jpn = {j = (j1 , j2 , · · · , jn ) ∈ Zn | −2p ≤ ji ≤ 2p, 1 ≤ i ≤ n}
and cp,q,i = kgIp,q k22,i listed above. Then there exists a constant c, cp,q such that for all
f ∈ l1 (Zn ) ∩ l2 (Zn ),
∑ ∑ p,q
cj f (m)f (m + j)
j∈Jpn m∈Zn
≤cγ(S̃ + Jpn , Σ)
(
∑
m∈Zn \(S+Jpn )
where cp,q
j =
∏n
i=1 cp,q,ji
∫
|f (m)|2 +
∫
Tn \Σ
)
|fˇ(λ)|2 |λ|10 dλ ,
|fˇ(λ)|2 dλ + cp,q
Σ
and γ(S̃ + Jpn , Σ) is the Nazarov constant.
References
[1] P. Jaming, Nazarov’s uncertainty principles in higher dimension, arXiv:math CA/0612367DOI
10-1016/j:jat 200704005, 1–10.
[2] F. Nazarov, Local estimates for exponential polynomials and their applications to inequalities of
the uncertainty principle type, Algebra i Analiz 5;4 (1993), 3–66, Translated in St. Petersburg.
Math. J. 5:4 (1994), 663–717.
[3] J. F. Price and P. C. Racki, Local uncertainty inequalities for Fourier series, Proc. Amer.
Math. Soc. 93 (1985), 245–251.
Communicated by Eiichi Nakai
Ahmed Abouelaz
Department of Mathematics and Computer Science,
Faculty of Sciences Aı̈n Chock,
University Hassan II of Casablanca,
Route d’El Jadida, Km 8, B.P. 5366 Maârif,
20100 Casablanca, Morocco
email: [email protected]
284
A. ABOUELAZ AND T. KAWAZOE
Takeshi Kawazoe
Department of Mathematics,
Keio University at Fujisawa,
Endo, Fujisawa, Kanagawa, 252-8520, Japan
email: [email protected]